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Some textbooks on algebraic topology define a simplicial complex as a special kind of subset of a Euclidean space (e.g., A. Wallace, p.3). Others define an abstract simplicial complex as a special kind of combinatorial structure (e.g., E. H. Spanier, p. 108). There exists a functor from the category of abstract simplicial complexes and simplicial maps to the category of topological spaces and continuous functions. My question is, what is an algebraic criterion for deciding whether or not there exists an embedding of the realization of an abstract simplicial complex into a Euclidean space of specified dimension. For example (J. R. Munkres, p. 18) exhibits an abstract simplicial complex whose realization is a Klein bottle, which cannot be embedded in $R^3$.

If a concrete 2-dimensional finite simplicial complex consists of point, line segment, and triangular subsets of 3-dimensional Euclidean space, then obviously there is more information about these simplices and their relationships than is contained in the corresponding abstract simplicial complex. In particular, there are the lengths of the line segments, the angles between connected line segments (one shared point), and the dihedral angles between connected triangles (one shared edge). This is a finite collection of data. My actual motivation for the titular question is the following question. Given an abstract finite simplicial complex together with "lengths" of 1-dimensional abstract simplices, "angles" between connected 1-dimensional abstract simplices, and "dihedral angles" between connected 2-dimensional abstract simplices, is there an algorithm to decide whether there is an embedding of its simplicial realization in 3-dimensional Euclidean space?

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    $\begingroup$ The paper Embedding obstructions in Rd from the Goodwillie-Weiss calculus and Whitney disks by Arone and Krushkal (arxiv.org/abs/2101.10995 - from today!) has examples of obstructions to embedding certain $2$-complexes into $\mathbb R^4$. Since every smooth manifold is homeomorphic to an abstract simplicial complex, it would help what precisely you mean by "algebraic". $\endgroup$ Jan 27, 2021 at 22:50
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    $\begingroup$ Let's back up. Are you aware of the Whitney embedding theorem, which gives a good partial answer to the less general problem of embedding manifolds into Euclidean space? Abstract simplicial complexes should be thought of as a very broad class of spaces from an algebraic topology perspective. For finite complexes, you can get crude bounds based on the number of simplices. But for anything more refined, I think you'll have to restrict attention to more specific classes of simplicial complexes. $\endgroup$
    – Tim Campion
    Jan 27, 2021 at 23:15
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    $\begingroup$ No criterion is known that answers your question in the stated generality. Moreover, any such criterion would almost certainly be difficult to impossible to compute in the "interesting" cases. But if the co-dimension is high, such embeddings are known to always exist. It's the low co-dimension case that is difficult. $\endgroup$ Jan 28, 2021 at 0:53
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    $\begingroup$ As evidence that this is a very difficult problem, Don Davis maintains a list of results about real projective space--known upper bounds for dimensions that they can't embed in, known lower bounds for dimensions that they can embed in, who the result is due to--that is quite extensive. lehigh.edu/~dmd1/imms.html $\endgroup$ Jan 28, 2021 at 14:55
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    $\begingroup$ Not necessarily useful for your purposes, but just to record this here: an abstract simplicial complex consisting of subsets of $\{1,2,\ldots,n\}$ can be embedded in $\mathbb{R}^n$. $\endgroup$ Jan 29, 2021 at 18:27

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It is not exactly clear to me what you mean by an algebraic criterion. But Uli Wagner and Martin Tancer have, with various coauthors and in several papers, tried to give an answer to the question of deciding embeddability, see for instance Filakovský, Wagner, and Zhechev - Embeddability of Simplicial Complexes is Undecidable or de Mesmay, Rieck, Sedgwick, and Tancer - Embeddability in $\mathbb R^3$ is NP-hard.

Of course, they are more interested in the complexity aspect, but an intermediate step is always finding a criterion for embeddability, and then checking it.

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  • $\begingroup$ I think that the ACM Mesmay et al paper says EMBED(2->3) is decidable and NP-hard, and that is closely related to my actual interest, which is provided in a new edit to the original question. $\endgroup$ Jan 29, 2021 at 18:00

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